We resume our discussion of the operator \(T_\alpha \) . It was shown by Börjeson [1]} that \(\Vert T_\alpha \Vert _{L^p({\mathbb {R}}^n)\rightarrow L^q({\mathbb {R}}^n)}<\infty \)  only if \((\frac{1}{p}, \frac{1}{q})\)  lies in the set
\(\mathcal {P}_{\alpha }^{n+1}&:= [B_{\alpha }^{n+1}, (B_{\alpha }^{n+1})^{\prime } , D_{\alpha }^{n+1}, (D_{\alpha }^{n+1})^{\prime }, H] \setminus \big ( [B_{\alpha }^{n+1}, D_{\alpha }^{n+1}] \cup [B_{\alpha }^{n+1}, D_{\alpha }^{n+1}]^{\prime } \big )\\& = \big \lbrace (x,y)\in \mathbf {Q} \colon x-y\ge \tfrac{2\alpha }{n+1}, \, x>\tfrac{n-1+2\alpha }{2n}, \, y < \tfrac{n+1-2\alpha }{2n} \big \rbrace .\) 
